Calculus Exam I Review: Limits and Derivatives – Step-by-Step Guidance

Q2. The graph of $y = f(x)$ is shown above. Find the following limits:

Background

Topic: Limits from a Graph

This question tests your ability to interpret a graph and determine the value of a function's limit at various points, including at infinity and at points of discontinuity. You will need to analyze the behavior of the function as $x$ approaches specific values from the left and right, as well as as $x$ approaches infinity or negative infinity.

Key Terms and Formulas:

• Limit: The value that a function approaches as the input approaches a certain value.

• One-sided limits: $\lim_{x \to c^-} f(x)$ (from the left), $\lim_{x \to c^+} f(x)$ (from the right)

• Limit at infinity: $\lim_{x \to \infty} f(x)$ or $\lim_{x \to -\infty} f(x)$

• Discontinuity: A point where the function is not continuous (jump, removable, or infinite).

Step-by-Step Guidance

1. Carefully examine the graph at each point of interest (e.g., $x = 4$, $x = -2$, $x = 9$, $x \to \infty$, $x \to -\infty$). Look for open circles, jumps, or vertical asymptotes that indicate discontinuities.

2. For each limit, determine whether you need to consider the left-hand limit, right-hand limit, or both. For example, $\lim_{x \to 4^-} f(x)$ means approaching $x = 4$ from the left.

3. Read the $y$-value that the function approaches as $x$ gets close to the target value. If the left and right limits are not equal, the two-sided limit does not exist at that point.

4. For limits at infinity, observe the end behavior of the graph as $x$ becomes very large or very small. Does the function approach a horizontal asymptote, or does it increase/decrease without bound?

5. If the graph shows a vertical asymptote, check whether the function approaches $\infty$ or $-\infty$ as $x$ approaches the asymptote from either side.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $\lim_{x \to -\infty} f(x) = \infty$

• (b) $\lim_{x \to \infty} f(x) = 6$

• (c) $\lim_{x \to 4^-} f(x) = 4$

• (d) $\lim_{x \to 4^+} f(x) = 6$

• (e) $\lim_{x \to 4} f(x)$ does not exist (DNE), since the left and right limits are not equal.

• (f) $\lim_{x \to -2} f(x) = 1$

• (g) $\lim_{x \to -4} f(x) = 4$

• (h) $\lim_{x \to 9} f(x) = -\infty$

Each answer is based on the behavior of the graph at the specified $x$-values. Discontinuities and asymptotes are identified by open circles, jumps, or arrows indicating infinite behavior.